Lower Bound for Independence Covering in $C_4$-Free Graphs

TL;DR

This paper proves that $C_4$-free graphs (or $K_{2,2}$-free graphs) cannot have small $k$-independence covering families of size polynomial in $n$, showing limitations in extending previous results to broader graph classes.

Contribution

It establishes a lower bound demonstrating that $K_{2,2}$-free graphs do not admit small $k$-independence covering families, contrasting with known results for degenerate graphs.

Findings

$K_{2,2}$-free graphs do not admit small $k$-independence covering families.

Lower bound of size $f(k)n^{k/4- ext{epsilon}}$ for such families.

Limits the applicability of previous covering family constructions to $C_4$-free graphs.

Abstract

An independent set in a graph $G$ is a set $S$ of pairwise non-adjacent vertices in $G$. A family $\mathcal{F}$ of independent sets in $G$ is called a $k$-independence covering family if for every independent set $I$ in $G$ of size at most $k$, there exists an $S \in \mathcal{F}$ such that $I \subseteq S$. Lokshtanov et al. [ACM Transactions on Algorithms, 2018] showed that graphs of degeneracy $d$ admit $k$-independence covering families of size $\binom{k(d+1)}{k} \cdot 2^{o(kd)} \cdot \log n$, and used this result to design efficient parameterized algorithms for a number of problems, including STABLE ODD CYCLE TRANSVERSAL and STABLE MULTICUT. In light of the results of Lokshtanov et al. it is quite natural to ask whether even more general families of graphs admit $k$-independence covering families of size $f(k)n^{O(1)}$. Graphs that exclude a complete bipartite graph $K_{d+1,d+1}$ with $d+1$ vertices on both sides as a subgraph, called $K_{d+1,d+1}$-free graphs, are a frequently considered generalization of $d$-degenerate graphs. This motivates the question whether $K_{d,d}$-free graphs admit $k$-independence covering families of size $f(k,d)n^{O(1)}$. Our main result is a resounding "no" to this question -- specifically we prove that even $K_{2,2}$-free graphs (or equivalently $C_4$-free graphs) do not admit $k$-independence covering families of size $f(k)n^{\frac{k}{4}-\epsilon}$.